scholarly journals Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Sixian Jin ◽  
Henry Schellhorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Malliavin Calculus ◽  
Series Representation ◽  
Parabolic Type ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Path Dependent

We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.

scholarly journals Modification of balayage spaces by transitions with application to coupling of PDE’s

2003 ◽  
Vol 169 ◽  
pp. 77-118 ◽  
Author(s):  
Wolfhard Hansen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential

AbstractModifications of balayage spaces are studied which, in probabilistic terms, correspond to killing and transitions (creation of mass combined with jumps). This is achieved by a modification of harmonic kernels for sufficiently small open sets. Applications to coupling of elliptic and parabolic partial differential equations of second order are discussed.

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